one crazy passenger

A line of 100 airline passengers is waiting to board a plane. They each hold a ticket to one of the 100 seats on that flight. Unfortunately, the first person in line is crazy, and will ignore the seat number on their ticket, picking a random seat to occupy. All of the other passengers are quite normal, and will go to their proper seat unless it is already occupied. If it is occupied, they will then find a free seat to sit in, at random. What is the probability that the last person to board the plane will sit in their proper seat?

Will the probability change if you add another person? Why or why not?

Originally posted by tomtom232
A line of 100 airline passengers is waiting to board a plane. They each hold a ticket to one of the 100 seats on that flight. Unfortunately, the first person in line is crazy, and will ignore the seat number on their ticket, picking a random seat to occupy. All of the other passengers are quite normal, and will go to their proper seat unless it is already ...[text shortened]... in their proper seat?

Will the probability change if you add another person? Why or why not?

At first I was thinking 1/100, but then I realized that only two seats on the plane really matter, seat 1 and seat 100. For every person that is displaced (and it doesn't matter how many), each displaced person has an equal probability of sitting in either seat 1 or seat 100.

If anyone sits in seat 1, everyone else will sit in the correct seat. [100%]
On the other hand, if anyone sits in seat 100, the last person can't sit in their own seat [0%].

Therefore, I would think that the average of these, 50%, is the probability of the last person sitting in their own seat.

If this reasoning holds, the the number of passengers is irrelevant.